We analyze a sample of galaxies with stellar masses greater than $10^{10} M_{\odot}$ and with redshifts in the range $0.025<z<0.05$ for which HI mass measurements are available from the GALEX Arecibo SDSS Survey (GASS) or from the Arecibo Legacy Fast ALFA survey (ALFALFA). At a given value of $M_*$, our sample consists primarily of galaxies that are more HI-rich than average. We constructed a series of three control samples for comparison with these HI-rich galaxies. As expected, HI-rich galaxies differ strongly from galaxies of same stellar mass that are selected without regard to HI content. The majority of these differences are attributable to the fact that galaxies with more gas are bluer and more actively star-forming. In order to identify those galaxy properties that are causally connected with HI content, we compare results derived for the HI sample with those derived for galaxies matched in stellar mass, size and NUV-$r$ colour. The only photometric property that is clearly attributable to increasing HI content, is the colour gradient of the galaxy. Galaxies with larger HI fractions have bluer, more actively star-forming outer disks compared to the inner part of the galaxy. HI-rich galaxies also have larger $g$-band radii compared to $i$-band radii. Our results are consistent with the 'inside-out' picture of disk galaxy formation, which has commonly served as a basis for semi-analytic models of the formation of disks in the context of Cold Dark Matter cosmologies. The lack of any intrinsic connection between HI fraction and galaxy asymmetry suggests that gas is accreted smoothly onto the outer disk.